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Asymptotic formulae for generalized Freud polynomials. (English) Zbl 1295.30014
Summary: We establish Mehler-Heine type formulae for orthonormal polynomials with respect to generalized Freud weights. Using this type of asymptotics, we can give estimates of the value at the origin of these polynomials and of all their derivatives as well as the asymptotic behavior of the corresponding zeros.

MSC:
30C10 Polynomials and rational functions of one complex variable
30E15 Asymptotic representations in the complex plane
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] Alfaro, M.; Marcellán, F.; Rezola, M. L.; Ronveaux, A., On orthogonal polynomials of Sobolev type: algebraic properties and zeros, SIAM J. Math. Anal., 23, 737-757, (1992) · Zbl 0764.33003
[2] Ganzburg, M., Limit theorems of polynomial approximation with exponential weights, Mem. Amer. Math. Soc., 192, 897, (2008) · Zbl 1142.30011
[3] Guadalupe, J. J.; Pérez, M.; Ruiz, F. J.; Varona, J. L., Asymptotic behaviour of orthogonal polynomials relative to measures with mass points, Mathematika, 40, 331-344, (1993) · Zbl 0791.42016
[4] Jung, H. S.; Sakai, R., Orthonormal polynomials with exponential-type weights, J. Approx. Theory, 152, 215-238, (2008) · Zbl 1152.41001
[5] Kriecherbauer, T.; McLaughlin, K. T.-R., Strong asymptotics of polynomials orthogonal with respect to freud weights, Int. Math. Res. Not., 6, 299-333, (1999) · Zbl 0944.42014
[6] Levin, A. L.; Lubinsky, D. S., Orthogonal polynomials for exponential weights, CMS Books Math., vol. 4, (2001), Springer-Verlag New York · Zbl 0997.42011
[7] Mhaskar, H. N.; Saff, E. B., Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc., 285, 203-234, (1984) · Zbl 0546.41014
[8] Nevai, P., Asymptotics for orthogonal polynomials associated with \(\exp(- x^4)\), SIAM J. Math. Anal., 15, 1177-1187, (1984) · Zbl 0566.42016
[9] Saff, E. B.; Totik, V., Logarithmic potentials with external fields, Grundlehren Math. Wiss., vol. 316, (1997), Springer-Verlag Berlin · Zbl 0881.31001
[10] Sheen, R. C., Plancherel-rotach type asymptotics for orthogonal polynomials associated with \(\exp(- x^6 / 6)\), J. Approx. Theory, 50, 232-293, (1987) · Zbl 0617.42017
[11] Szegő, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, (1975), Amer. Math. Soc. Providence, RI · JFM 61.0386.03
[12] Temme, N. M., Special functions. an introduction to the classical functions of mathematical physics, (1996), Wiley New York · Zbl 0856.33001
[13] Totik, V., Orthogonal polynomials, Surv. Approx. Theory, 1, 70-125, (2005) · Zbl 1105.42017
[14] Wong, R.; Zhang, L., Global asymptotics of orthogonal polynomials associated with \(| x |^{2 \alpha} e^{- Q(x)}\), J. Approx. Theory, 162, 723-765, (2010) · Zbl 1198.33004
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